Dirichlet series
| L(s) = 1 | − 2·2-s + 2.94e4·3-s + 2·4-s − 7.30e6·5-s − 5.88e4·6-s − 5.85e8·7-s + 3.09e8·8-s + 4.33e8·9-s + 1.46e7·10-s − 6.50e9·11-s + 5.88e4·12-s − 3.69e11·13-s + 1.17e9·14-s − 2.15e11·15-s + 1.14e12·16-s + 6.50e12·17-s − 8.67e8·18-s − 1.46e7·20-s − 1.72e13·21-s + 1.30e10·22-s − 8.16e13·23-s + 9.10e12·24-s + 4.36e13·25-s + 7.38e11·26-s − 1.32e14·27-s − 1.17e9·28-s + 4.30e11·30-s + ⋯ |
| L(s) = 1 | − 0.00195·2-s + 0.498·3-s + 1.90e−6·4-s − 0.747·5-s − 0.000974·6-s − 2.07·7-s + 0.288·8-s + 0.124·9-s + 0.00146·10-s − 0.250·11-s − 2.67·13-s + 0.00404·14-s − 0.372·15-s + 1.04·16-s + 3.22·17-s − 0.000242·18-s − 1.42e − 6·20-s − 1.03·21-s + 0.000489·22-s − 1.97·23-s + 0.143·24-s + 0.457·25-s + 0.00523·26-s − 0.641·27-s − 3.95e − 6·28-s + 0.000728·30-s + 2.12·31-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(21-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{18}\right)^{s/2} \, \Gamma_{\C}(s+10)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(5^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.13654\times 10^{19}\) |
| Root analytic conductor: | \(3.56029\) |
| Motivic weight: | \(20\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 5^{18} ,\ ( \ : [10]^{18} ),\ 1 )\) |
Particular Values
| \(L(\frac{21}{2})\) | \(\approx\) | \(0.06117323910\) |
| \(L(\frac12)\) | \(\approx\) | \(0.06117323910\) |
| \(L(11)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 + 1460428 p T + 77631858937 p^{3} T^{2} + 30803189585954848 p^{7} T^{3} + \)\(12\!\cdots\!64\)\( p^{12} T^{4} + \)\(13\!\cdots\!44\)\( p^{17} T^{5} + \)\(24\!\cdots\!52\)\( p^{23} T^{6} + \)\(45\!\cdots\!52\)\( p^{30} T^{7} + \)\(17\!\cdots\!88\)\( p^{39} T^{8} + \)\(57\!\cdots\!16\)\( p^{51} T^{9} + \)\(17\!\cdots\!88\)\( p^{59} T^{10} + \)\(45\!\cdots\!52\)\( p^{70} T^{11} + \)\(24\!\cdots\!52\)\( p^{83} T^{12} + \)\(13\!\cdots\!44\)\( p^{97} T^{13} + \)\(12\!\cdots\!64\)\( p^{112} T^{14} + 30803189585954848 p^{127} T^{15} + 77631858937 p^{143} T^{16} + 1460428 p^{161} T^{17} + p^{180} T^{18} \) |
| good | 2 | \( 1 + p T + p T^{2} - 9666839 p^{5} T^{3} - 17941008763 p^{6} T^{4} + 7869555366553 p^{7} T^{5} + 389548156749553 p^{7} T^{6} - 572443527526212467 p^{11} T^{7} + \)\(11\!\cdots\!05\)\( p^{14} T^{8} - \)\(11\!\cdots\!97\)\( p^{21} T^{9} + \)\(11\!\cdots\!07\)\( p^{23} T^{10} + \)\(17\!\cdots\!13\)\( p^{29} T^{11} - \)\(21\!\cdots\!47\)\( p^{30} T^{12} + \)\(27\!\cdots\!91\)\( p^{35} T^{13} + \)\(25\!\cdots\!01\)\( p^{39} T^{14} - \)\(35\!\cdots\!77\)\( p^{48} T^{15} + \)\(52\!\cdots\!09\)\( p^{52} T^{16} + \)\(77\!\cdots\!95\)\( p^{57} T^{17} + \)\(47\!\cdots\!45\)\( p^{61} T^{18} + \)\(77\!\cdots\!95\)\( p^{77} T^{19} + \)\(52\!\cdots\!09\)\( p^{92} T^{20} - \)\(35\!\cdots\!77\)\( p^{108} T^{21} + \)\(25\!\cdots\!01\)\( p^{119} T^{22} + \)\(27\!\cdots\!91\)\( p^{135} T^{23} - \)\(21\!\cdots\!47\)\( p^{150} T^{24} + \)\(17\!\cdots\!13\)\( p^{169} T^{25} + \)\(11\!\cdots\!07\)\( p^{183} T^{26} - \)\(11\!\cdots\!97\)\( p^{201} T^{27} + \)\(11\!\cdots\!05\)\( p^{214} T^{28} - 572443527526212467 p^{231} T^{29} + 389548156749553 p^{247} T^{30} + 7869555366553 p^{267} T^{31} - 17941008763 p^{286} T^{32} - 9666839 p^{305} T^{33} + p^{321} T^{34} + p^{341} T^{35} + p^{360} T^{36} \) |
| 3 | \( 1 - 3272 p^{2} T + 5352992 p^{4} T^{2} + 1631637960992 p^{4} T^{3} - 3444476020644418523 p^{2} T^{4} + \)\(78\!\cdots\!64\)\( p^{4} T^{5} + \)\(45\!\cdots\!96\)\( p^{6} T^{6} - \)\(25\!\cdots\!56\)\( p^{8} T^{7} + \)\(25\!\cdots\!80\)\( p^{10} T^{8} - \)\(27\!\cdots\!28\)\( p^{13} T^{9} + \)\(70\!\cdots\!36\)\( p^{16} T^{10} - \)\(14\!\cdots\!32\)\( p^{19} T^{11} + \)\(10\!\cdots\!08\)\( p^{22} T^{12} - \)\(42\!\cdots\!28\)\( p^{26} T^{13} + \)\(55\!\cdots\!12\)\( p^{30} T^{14} - \)\(76\!\cdots\!44\)\( p^{33} T^{15} - \)\(20\!\cdots\!66\)\( p^{36} T^{16} + \)\(14\!\cdots\!40\)\( p^{40} T^{17} - \)\(67\!\cdots\!60\)\( p^{44} T^{18} + \)\(14\!\cdots\!40\)\( p^{60} T^{19} - \)\(20\!\cdots\!66\)\( p^{76} T^{20} - \)\(76\!\cdots\!44\)\( p^{93} T^{21} + \)\(55\!\cdots\!12\)\( p^{110} T^{22} - \)\(42\!\cdots\!28\)\( p^{126} T^{23} + \)\(10\!\cdots\!08\)\( p^{142} T^{24} - \)\(14\!\cdots\!32\)\( p^{159} T^{25} + \)\(70\!\cdots\!36\)\( p^{176} T^{26} - \)\(27\!\cdots\!28\)\( p^{193} T^{27} + \)\(25\!\cdots\!80\)\( p^{210} T^{28} - \)\(25\!\cdots\!56\)\( p^{228} T^{29} + \)\(45\!\cdots\!96\)\( p^{246} T^{30} + \)\(78\!\cdots\!64\)\( p^{264} T^{31} - 3444476020644418523 p^{282} T^{32} + 1631637960992 p^{304} T^{33} + 5352992 p^{324} T^{34} - 3272 p^{342} T^{35} + p^{360} T^{36} \) | |
| 7 | \( 1 + 11949648 p^{2} T + 71397043661952 p^{4} T^{2} + \)\(22\!\cdots\!64\)\( p^{3} T^{3} + \)\(37\!\cdots\!57\)\( p^{2} T^{4} - \)\(70\!\cdots\!16\)\( p^{4} T^{5} - \)\(22\!\cdots\!84\)\( p^{6} T^{6} - \)\(13\!\cdots\!88\)\( p^{5} T^{7} - \)\(49\!\cdots\!80\)\( p^{4} T^{8} - \)\(17\!\cdots\!08\)\( p^{7} T^{9} - \)\(47\!\cdots\!44\)\( p^{8} T^{10} - \)\(81\!\cdots\!08\)\( p^{7} T^{11} - \)\(16\!\cdots\!72\)\( p^{6} T^{12} - \)\(12\!\cdots\!12\)\( p^{8} T^{13} - \)\(49\!\cdots\!88\)\( p^{10} T^{14} - \)\(39\!\cdots\!12\)\( p^{12} T^{15} + \)\(55\!\cdots\!86\)\( p^{14} T^{16} + \)\(66\!\cdots\!40\)\( p^{16} T^{17} + \)\(31\!\cdots\!60\)\( p^{18} T^{18} + \)\(66\!\cdots\!40\)\( p^{36} T^{19} + \)\(55\!\cdots\!86\)\( p^{54} T^{20} - \)\(39\!\cdots\!12\)\( p^{72} T^{21} - \)\(49\!\cdots\!88\)\( p^{90} T^{22} - \)\(12\!\cdots\!12\)\( p^{108} T^{23} - \)\(16\!\cdots\!72\)\( p^{126} T^{24} - \)\(81\!\cdots\!08\)\( p^{147} T^{25} - \)\(47\!\cdots\!44\)\( p^{168} T^{26} - \)\(17\!\cdots\!08\)\( p^{187} T^{27} - \)\(49\!\cdots\!80\)\( p^{204} T^{28} - \)\(13\!\cdots\!88\)\( p^{225} T^{29} - \)\(22\!\cdots\!84\)\( p^{246} T^{30} - \)\(70\!\cdots\!16\)\( p^{264} T^{31} + \)\(37\!\cdots\!57\)\( p^{282} T^{32} + \)\(22\!\cdots\!64\)\( p^{303} T^{33} + 71397043661952 p^{324} T^{34} + 11949648 p^{342} T^{35} + p^{360} T^{36} \) | |
| 11 | \( ( 1 + 3253171532 T + \)\(34\!\cdots\!53\)\( T^{2} + \)\(75\!\cdots\!84\)\( T^{3} + \)\(64\!\cdots\!60\)\( T^{4} + \)\(10\!\cdots\!12\)\( p T^{5} + \)\(66\!\cdots\!84\)\( p^{2} T^{6} + \)\(83\!\cdots\!96\)\( p^{3} T^{7} + \)\(49\!\cdots\!38\)\( p^{4} T^{8} + \)\(53\!\cdots\!80\)\( p^{5} T^{9} + \)\(49\!\cdots\!38\)\( p^{24} T^{10} + \)\(83\!\cdots\!96\)\( p^{43} T^{11} + \)\(66\!\cdots\!84\)\( p^{62} T^{12} + \)\(10\!\cdots\!12\)\( p^{81} T^{13} + \)\(64\!\cdots\!60\)\( p^{100} T^{14} + \)\(75\!\cdots\!84\)\( p^{120} T^{15} + \)\(34\!\cdots\!53\)\( p^{140} T^{16} + 3253171532 p^{160} T^{17} + p^{180} T^{18} )^{2} \) | |
| 13 | \( 1 + 369354655602 T + \)\(68\!\cdots\!02\)\( T^{2} + \)\(11\!\cdots\!02\)\( T^{3} + \)\(17\!\cdots\!93\)\( T^{4} + \)\(17\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!84\)\( T^{6} + \)\(27\!\cdots\!36\)\( p^{2} T^{7} - \)\(15\!\cdots\!40\)\( p^{3} T^{8} - \)\(50\!\cdots\!52\)\( p^{3} T^{9} - \)\(66\!\cdots\!04\)\( p^{4} T^{10} - \)\(74\!\cdots\!08\)\( p^{5} T^{11} - \)\(72\!\cdots\!92\)\( p^{6} T^{12} - \)\(47\!\cdots\!36\)\( p^{7} T^{13} - \)\(14\!\cdots\!72\)\( p^{8} T^{14} + \)\(21\!\cdots\!56\)\( p^{9} T^{15} + \)\(46\!\cdots\!94\)\( p^{12} T^{16} + \)\(99\!\cdots\!40\)\( p^{12} T^{17} + \)\(14\!\cdots\!40\)\( p^{12} T^{18} + \)\(99\!\cdots\!40\)\( p^{32} T^{19} + \)\(46\!\cdots\!94\)\( p^{52} T^{20} + \)\(21\!\cdots\!56\)\( p^{69} T^{21} - \)\(14\!\cdots\!72\)\( p^{88} T^{22} - \)\(47\!\cdots\!36\)\( p^{107} T^{23} - \)\(72\!\cdots\!92\)\( p^{126} T^{24} - \)\(74\!\cdots\!08\)\( p^{145} T^{25} - \)\(66\!\cdots\!04\)\( p^{164} T^{26} - \)\(50\!\cdots\!52\)\( p^{183} T^{27} - \)\(15\!\cdots\!40\)\( p^{203} T^{28} + \)\(27\!\cdots\!36\)\( p^{222} T^{29} + \)\(13\!\cdots\!84\)\( p^{240} T^{30} + \)\(17\!\cdots\!84\)\( p^{260} T^{31} + \)\(17\!\cdots\!93\)\( p^{280} T^{32} + \)\(11\!\cdots\!02\)\( p^{300} T^{33} + \)\(68\!\cdots\!02\)\( p^{320} T^{34} + 369354655602 p^{340} T^{35} + p^{360} T^{36} \) | |
| 17 | \( 1 - 6508314764998 T + \)\(21\!\cdots\!02\)\( T^{2} - \)\(35\!\cdots\!94\)\( p T^{3} + \)\(57\!\cdots\!37\)\( p^{2} T^{4} - \)\(37\!\cdots\!96\)\( p^{4} T^{5} + \)\(44\!\cdots\!04\)\( p^{4} T^{6} - \)\(12\!\cdots\!88\)\( p^{5} T^{7} - \)\(70\!\cdots\!20\)\( p^{6} T^{8} + \)\(21\!\cdots\!72\)\( p^{7} T^{9} - \)\(35\!\cdots\!84\)\( p^{8} T^{10} + \)\(46\!\cdots\!48\)\( p^{9} T^{11} - \)\(54\!\cdots\!72\)\( p^{10} T^{12} + \)\(45\!\cdots\!36\)\( p^{11} T^{13} - \)\(15\!\cdots\!92\)\( p^{12} T^{14} - \)\(29\!\cdots\!76\)\( p^{13} T^{15} + \)\(89\!\cdots\!66\)\( p^{14} T^{16} - \)\(15\!\cdots\!20\)\( p^{15} T^{17} + \)\(20\!\cdots\!40\)\( p^{16} T^{18} - \)\(15\!\cdots\!20\)\( p^{35} T^{19} + \)\(89\!\cdots\!66\)\( p^{54} T^{20} - \)\(29\!\cdots\!76\)\( p^{73} T^{21} - \)\(15\!\cdots\!92\)\( p^{92} T^{22} + \)\(45\!\cdots\!36\)\( p^{111} T^{23} - \)\(54\!\cdots\!72\)\( p^{130} T^{24} + \)\(46\!\cdots\!48\)\( p^{149} T^{25} - \)\(35\!\cdots\!84\)\( p^{168} T^{26} + \)\(21\!\cdots\!72\)\( p^{187} T^{27} - \)\(70\!\cdots\!20\)\( p^{206} T^{28} - \)\(12\!\cdots\!88\)\( p^{225} T^{29} + \)\(44\!\cdots\!04\)\( p^{244} T^{30} - \)\(37\!\cdots\!96\)\( p^{264} T^{31} + \)\(57\!\cdots\!37\)\( p^{282} T^{32} - \)\(35\!\cdots\!94\)\( p^{301} T^{33} + \)\(21\!\cdots\!02\)\( p^{320} T^{34} - 6508314764998 p^{340} T^{35} + p^{360} T^{36} \) | |
| 19 | \( 1 - \)\(12\!\cdots\!38\)\( p^{2} T^{2} + \)\(37\!\cdots\!47\)\( p^{5} T^{4} - \)\(26\!\cdots\!36\)\( p^{6} T^{6} + \)\(69\!\cdots\!60\)\( p^{8} T^{8} - \)\(13\!\cdots\!68\)\( p^{10} T^{10} + \)\(20\!\cdots\!24\)\( p^{12} T^{12} - \)\(25\!\cdots\!44\)\( p^{14} T^{14} + \)\(28\!\cdots\!18\)\( p^{16} T^{16} - \)\(22\!\cdots\!20\)\( p^{22} T^{18} + \)\(28\!\cdots\!18\)\( p^{56} T^{20} - \)\(25\!\cdots\!44\)\( p^{94} T^{22} + \)\(20\!\cdots\!24\)\( p^{132} T^{24} - \)\(13\!\cdots\!68\)\( p^{170} T^{26} + \)\(69\!\cdots\!60\)\( p^{208} T^{28} - \)\(26\!\cdots\!36\)\( p^{246} T^{30} + \)\(37\!\cdots\!47\)\( p^{285} T^{32} - \)\(12\!\cdots\!38\)\( p^{322} T^{34} + p^{360} T^{36} \) | |
| 23 | \( 1 + 81655963656152 T + \)\(33\!\cdots\!52\)\( T^{2} + \)\(18\!\cdots\!52\)\( T^{3} + \)\(16\!\cdots\!93\)\( T^{4} + \)\(82\!\cdots\!84\)\( T^{5} + \)\(27\!\cdots\!84\)\( T^{6} + \)\(12\!\cdots\!84\)\( T^{7} + \)\(65\!\cdots\!20\)\( T^{8} + \)\(20\!\cdots\!56\)\( T^{9} + \)\(44\!\cdots\!56\)\( T^{10} + \)\(47\!\cdots\!56\)\( T^{11} - \)\(47\!\cdots\!28\)\( T^{12} - \)\(43\!\cdots\!12\)\( T^{13} - \)\(22\!\cdots\!12\)\( T^{14} - \)\(68\!\cdots\!44\)\( p T^{15} - \)\(93\!\cdots\!86\)\( T^{16} - \)\(35\!\cdots\!60\)\( T^{17} - \)\(12\!\cdots\!60\)\( T^{18} - \)\(35\!\cdots\!60\)\( p^{20} T^{19} - \)\(93\!\cdots\!86\)\( p^{40} T^{20} - \)\(68\!\cdots\!44\)\( p^{61} T^{21} - \)\(22\!\cdots\!12\)\( p^{80} T^{22} - \)\(43\!\cdots\!12\)\( p^{100} T^{23} - \)\(47\!\cdots\!28\)\( p^{120} T^{24} + \)\(47\!\cdots\!56\)\( p^{140} T^{25} + \)\(44\!\cdots\!56\)\( p^{160} T^{26} + \)\(20\!\cdots\!56\)\( p^{180} T^{27} + \)\(65\!\cdots\!20\)\( p^{200} T^{28} + \)\(12\!\cdots\!84\)\( p^{220} T^{29} + \)\(27\!\cdots\!84\)\( p^{240} T^{30} + \)\(82\!\cdots\!84\)\( p^{260} T^{31} + \)\(16\!\cdots\!93\)\( p^{280} T^{32} + \)\(18\!\cdots\!52\)\( p^{300} T^{33} + \)\(33\!\cdots\!52\)\( p^{320} T^{34} + 81655963656152 p^{340} T^{35} + p^{360} T^{36} \) | |
| 29 | \( 1 - \)\(16\!\cdots\!18\)\( T^{2} + \)\(13\!\cdots\!53\)\( T^{4} - \)\(78\!\cdots\!16\)\( T^{6} + \)\(35\!\cdots\!60\)\( T^{8} - \)\(12\!\cdots\!68\)\( T^{10} + \)\(37\!\cdots\!64\)\( T^{12} - \)\(96\!\cdots\!24\)\( T^{14} + \)\(21\!\cdots\!58\)\( T^{16} - \)\(40\!\cdots\!20\)\( T^{18} + \)\(21\!\cdots\!58\)\( p^{40} T^{20} - \)\(96\!\cdots\!24\)\( p^{80} T^{22} + \)\(37\!\cdots\!64\)\( p^{120} T^{24} - \)\(12\!\cdots\!68\)\( p^{160} T^{26} + \)\(35\!\cdots\!60\)\( p^{200} T^{28} - \)\(78\!\cdots\!16\)\( p^{240} T^{30} + \)\(13\!\cdots\!53\)\( p^{280} T^{32} - \)\(16\!\cdots\!18\)\( p^{320} T^{34} + p^{360} T^{36} \) | |
| 31 | \( ( 1 - 871128448450368 T + \)\(36\!\cdots\!53\)\( T^{2} - \)\(24\!\cdots\!16\)\( T^{3} + \)\(59\!\cdots\!60\)\( T^{4} - \)\(32\!\cdots\!68\)\( T^{5} + \)\(64\!\cdots\!64\)\( T^{6} - \)\(28\!\cdots\!24\)\( T^{7} + \)\(52\!\cdots\!58\)\( T^{8} - \)\(20\!\cdots\!20\)\( T^{9} + \)\(52\!\cdots\!58\)\( p^{20} T^{10} - \)\(28\!\cdots\!24\)\( p^{40} T^{11} + \)\(64\!\cdots\!64\)\( p^{60} T^{12} - \)\(32\!\cdots\!68\)\( p^{80} T^{13} + \)\(59\!\cdots\!60\)\( p^{100} T^{14} - \)\(24\!\cdots\!16\)\( p^{120} T^{15} + \)\(36\!\cdots\!53\)\( p^{140} T^{16} - 871128448450368 p^{160} T^{17} + p^{180} T^{18} )^{2} \) | |
| 37 | \( 1 - 6383289895587498 T + \)\(20\!\cdots\!02\)\( T^{2} - \)\(13\!\cdots\!98\)\( T^{3} + \)\(43\!\cdots\!93\)\( T^{4} - \)\(44\!\cdots\!16\)\( T^{5} + \)\(19\!\cdots\!84\)\( T^{6} - \)\(17\!\cdots\!16\)\( T^{7} + \)\(50\!\cdots\!20\)\( T^{8} - \)\(41\!\cdots\!44\)\( T^{9} + \)\(17\!\cdots\!56\)\( T^{10} + \)\(36\!\cdots\!56\)\( T^{11} - \)\(49\!\cdots\!28\)\( T^{12} - \)\(14\!\cdots\!12\)\( T^{13} + \)\(11\!\cdots\!88\)\( T^{14} - \)\(80\!\cdots\!12\)\( T^{15} - \)\(66\!\cdots\!86\)\( T^{16} - \)\(39\!\cdots\!60\)\( T^{17} + \)\(61\!\cdots\!40\)\( T^{18} - \)\(39\!\cdots\!60\)\( p^{20} T^{19} - \)\(66\!\cdots\!86\)\( p^{40} T^{20} - \)\(80\!\cdots\!12\)\( p^{60} T^{21} + \)\(11\!\cdots\!88\)\( p^{80} T^{22} - \)\(14\!\cdots\!12\)\( p^{100} T^{23} - \)\(49\!\cdots\!28\)\( p^{120} T^{24} + \)\(36\!\cdots\!56\)\( p^{140} T^{25} + \)\(17\!\cdots\!56\)\( p^{160} T^{26} - \)\(41\!\cdots\!44\)\( p^{180} T^{27} + \)\(50\!\cdots\!20\)\( p^{200} T^{28} - \)\(17\!\cdots\!16\)\( p^{220} T^{29} + \)\(19\!\cdots\!84\)\( p^{240} T^{30} - \)\(44\!\cdots\!16\)\( p^{260} T^{31} + \)\(43\!\cdots\!93\)\( p^{280} T^{32} - \)\(13\!\cdots\!98\)\( p^{300} T^{33} + \)\(20\!\cdots\!02\)\( p^{320} T^{34} - 6383289895587498 p^{340} T^{35} + p^{360} T^{36} \) | |
| 41 | \( ( 1 + 15301152298782332 T + \)\(99\!\cdots\!53\)\( T^{2} + \)\(11\!\cdots\!84\)\( T^{3} + \)\(42\!\cdots\!60\)\( T^{4} + \)\(35\!\cdots\!32\)\( T^{5} + \)\(99\!\cdots\!64\)\( T^{6} + \)\(55\!\cdots\!76\)\( T^{7} + \)\(17\!\cdots\!58\)\( T^{8} + \)\(73\!\cdots\!80\)\( T^{9} + \)\(17\!\cdots\!58\)\( p^{20} T^{10} + \)\(55\!\cdots\!76\)\( p^{40} T^{11} + \)\(99\!\cdots\!64\)\( p^{60} T^{12} + \)\(35\!\cdots\!32\)\( p^{80} T^{13} + \)\(42\!\cdots\!60\)\( p^{100} T^{14} + \)\(11\!\cdots\!84\)\( p^{120} T^{15} + \)\(99\!\cdots\!53\)\( p^{140} T^{16} + 15301152298782332 p^{160} T^{17} + p^{180} T^{18} )^{2} \) | |
| 43 | \( 1 + 46399469942287752 T + \)\(10\!\cdots\!52\)\( T^{2} + \)\(11\!\cdots\!52\)\( T^{3} + \)\(48\!\cdots\!93\)\( T^{4} + \)\(38\!\cdots\!84\)\( T^{5} + \)\(19\!\cdots\!84\)\( T^{6} + \)\(38\!\cdots\!84\)\( T^{7} + \)\(51\!\cdots\!20\)\( T^{8} + \)\(83\!\cdots\!56\)\( T^{9} + \)\(26\!\cdots\!56\)\( T^{10} + \)\(55\!\cdots\!56\)\( T^{11} + \)\(84\!\cdots\!72\)\( T^{12} + \)\(85\!\cdots\!88\)\( T^{13} + \)\(79\!\cdots\!88\)\( T^{14} - \)\(19\!\cdots\!12\)\( T^{15} + \)\(34\!\cdots\!14\)\( T^{16} + \)\(49\!\cdots\!40\)\( T^{17} + \)\(15\!\cdots\!40\)\( T^{18} + \)\(49\!\cdots\!40\)\( p^{20} T^{19} + \)\(34\!\cdots\!14\)\( p^{40} T^{20} - \)\(19\!\cdots\!12\)\( p^{60} T^{21} + \)\(79\!\cdots\!88\)\( p^{80} T^{22} + \)\(85\!\cdots\!88\)\( p^{100} T^{23} + \)\(84\!\cdots\!72\)\( p^{120} T^{24} + \)\(55\!\cdots\!56\)\( p^{140} T^{25} + \)\(26\!\cdots\!56\)\( p^{160} T^{26} + \)\(83\!\cdots\!56\)\( p^{180} T^{27} + \)\(51\!\cdots\!20\)\( p^{200} T^{28} + \)\(38\!\cdots\!84\)\( p^{220} T^{29} + \)\(19\!\cdots\!84\)\( p^{240} T^{30} + \)\(38\!\cdots\!84\)\( p^{260} T^{31} + \)\(48\!\cdots\!93\)\( p^{280} T^{32} + \)\(11\!\cdots\!52\)\( p^{300} T^{33} + \)\(10\!\cdots\!52\)\( p^{320} T^{34} + 46399469942287752 p^{340} T^{35} + p^{360} T^{36} \) | |
| 47 | \( 1 + 137109249757437752 T + \)\(93\!\cdots\!52\)\( T^{2} + \)\(41\!\cdots\!52\)\( T^{3} + \)\(72\!\cdots\!93\)\( T^{4} + \)\(18\!\cdots\!84\)\( T^{5} + \)\(43\!\cdots\!84\)\( T^{6} + \)\(45\!\cdots\!84\)\( T^{7} + \)\(37\!\cdots\!20\)\( T^{8} + \)\(24\!\cdots\!56\)\( T^{9} + \)\(13\!\cdots\!56\)\( T^{10} + \)\(75\!\cdots\!56\)\( T^{11} + \)\(31\!\cdots\!72\)\( T^{12} + \)\(11\!\cdots\!88\)\( T^{13} + \)\(57\!\cdots\!88\)\( T^{14} + \)\(39\!\cdots\!88\)\( T^{15} + \)\(32\!\cdots\!14\)\( T^{16} + \)\(22\!\cdots\!40\)\( T^{17} + \)\(12\!\cdots\!40\)\( T^{18} + \)\(22\!\cdots\!40\)\( p^{20} T^{19} + \)\(32\!\cdots\!14\)\( p^{40} T^{20} + \)\(39\!\cdots\!88\)\( p^{60} T^{21} + \)\(57\!\cdots\!88\)\( p^{80} T^{22} + \)\(11\!\cdots\!88\)\( p^{100} T^{23} + \)\(31\!\cdots\!72\)\( p^{120} T^{24} + \)\(75\!\cdots\!56\)\( p^{140} T^{25} + \)\(13\!\cdots\!56\)\( p^{160} T^{26} + \)\(24\!\cdots\!56\)\( p^{180} T^{27} + \)\(37\!\cdots\!20\)\( p^{200} T^{28} + \)\(45\!\cdots\!84\)\( p^{220} T^{29} + \)\(43\!\cdots\!84\)\( p^{240} T^{30} + \)\(18\!\cdots\!84\)\( p^{260} T^{31} + \)\(72\!\cdots\!93\)\( p^{280} T^{32} + \)\(41\!\cdots\!52\)\( p^{300} T^{33} + \)\(93\!\cdots\!52\)\( p^{320} T^{34} + 137109249757437752 p^{340} T^{35} + p^{360} T^{36} \) | |
| 53 | \( 1 + 184102027021671302 T + \)\(16\!\cdots\!02\)\( T^{2} - \)\(24\!\cdots\!98\)\( T^{3} + \)\(19\!\cdots\!93\)\( T^{4} + \)\(47\!\cdots\!84\)\( T^{5} + \)\(57\!\cdots\!84\)\( T^{6} - \)\(77\!\cdots\!16\)\( T^{7} + \)\(12\!\cdots\!20\)\( T^{8} + \)\(60\!\cdots\!56\)\( T^{9} + \)\(11\!\cdots\!56\)\( T^{10} - \)\(11\!\cdots\!44\)\( T^{11} - \)\(89\!\cdots\!28\)\( T^{12} + \)\(39\!\cdots\!88\)\( T^{13} + \)\(14\!\cdots\!88\)\( T^{14} - \)\(74\!\cdots\!12\)\( T^{15} - \)\(23\!\cdots\!86\)\( T^{16} + \)\(55\!\cdots\!40\)\( T^{17} + \)\(15\!\cdots\!40\)\( T^{18} + \)\(55\!\cdots\!40\)\( p^{20} T^{19} - \)\(23\!\cdots\!86\)\( p^{40} T^{20} - \)\(74\!\cdots\!12\)\( p^{60} T^{21} + \)\(14\!\cdots\!88\)\( p^{80} T^{22} + \)\(39\!\cdots\!88\)\( p^{100} T^{23} - \)\(89\!\cdots\!28\)\( p^{120} T^{24} - \)\(11\!\cdots\!44\)\( p^{140} T^{25} + \)\(11\!\cdots\!56\)\( p^{160} T^{26} + \)\(60\!\cdots\!56\)\( p^{180} T^{27} + \)\(12\!\cdots\!20\)\( p^{200} T^{28} - \)\(77\!\cdots\!16\)\( p^{220} T^{29} + \)\(57\!\cdots\!84\)\( p^{240} T^{30} + \)\(47\!\cdots\!84\)\( p^{260} T^{31} + \)\(19\!\cdots\!93\)\( p^{280} T^{32} - \)\(24\!\cdots\!98\)\( p^{300} T^{33} + \)\(16\!\cdots\!02\)\( p^{320} T^{34} + 184102027021671302 p^{340} T^{35} + p^{360} T^{36} \) | |
| 59 | \( 1 - \)\(20\!\cdots\!18\)\( T^{2} + \)\(20\!\cdots\!53\)\( T^{4} - \)\(13\!\cdots\!16\)\( T^{6} + \)\(69\!\cdots\!60\)\( T^{8} - \)\(29\!\cdots\!68\)\( T^{10} + \)\(10\!\cdots\!64\)\( T^{12} - \)\(36\!\cdots\!24\)\( T^{14} + \)\(10\!\cdots\!58\)\( T^{16} - \)\(29\!\cdots\!20\)\( T^{18} + \)\(10\!\cdots\!58\)\( p^{40} T^{20} - \)\(36\!\cdots\!24\)\( p^{80} T^{22} + \)\(10\!\cdots\!64\)\( p^{120} T^{24} - \)\(29\!\cdots\!68\)\( p^{160} T^{26} + \)\(69\!\cdots\!60\)\( p^{200} T^{28} - \)\(13\!\cdots\!16\)\( p^{240} T^{30} + \)\(20\!\cdots\!53\)\( p^{280} T^{32} - \)\(20\!\cdots\!18\)\( p^{320} T^{34} + p^{360} T^{36} \) | |
| 61 | \( ( 1 - 24939597015022068 T + \)\(29\!\cdots\!53\)\( T^{2} - \)\(29\!\cdots\!16\)\( T^{3} + \)\(40\!\cdots\!60\)\( T^{4} - \)\(53\!\cdots\!68\)\( T^{5} + \)\(37\!\cdots\!64\)\( T^{6} - \)\(46\!\cdots\!24\)\( T^{7} + \)\(24\!\cdots\!58\)\( T^{8} - \)\(26\!\cdots\!20\)\( T^{9} + \)\(24\!\cdots\!58\)\( p^{20} T^{10} - \)\(46\!\cdots\!24\)\( p^{40} T^{11} + \)\(37\!\cdots\!64\)\( p^{60} T^{12} - \)\(53\!\cdots\!68\)\( p^{80} T^{13} + \)\(40\!\cdots\!60\)\( p^{100} T^{14} - \)\(29\!\cdots\!16\)\( p^{120} T^{15} + \)\(29\!\cdots\!53\)\( p^{140} T^{16} - 24939597015022068 p^{160} T^{17} + p^{180} T^{18} )^{2} \) | |
| 67 | \( 1 + 5597336519263153752 T + \)\(15\!\cdots\!52\)\( T^{2} + \)\(36\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!93\)\( T^{4} + \)\(31\!\cdots\!84\)\( T^{5} + \)\(64\!\cdots\!84\)\( T^{6} + \)\(11\!\cdots\!84\)\( T^{7} + \)\(23\!\cdots\!20\)\( T^{8} + \)\(44\!\cdots\!56\)\( T^{9} + \)\(94\!\cdots\!68\)\( p T^{10} + \)\(61\!\cdots\!56\)\( T^{11} + \)\(22\!\cdots\!72\)\( T^{12} - \)\(12\!\cdots\!12\)\( T^{13} - \)\(54\!\cdots\!12\)\( T^{14} - \)\(17\!\cdots\!12\)\( T^{15} - \)\(41\!\cdots\!86\)\( T^{16} - \)\(81\!\cdots\!60\)\( T^{17} - \)\(14\!\cdots\!60\)\( T^{18} - \)\(81\!\cdots\!60\)\( p^{20} T^{19} - \)\(41\!\cdots\!86\)\( p^{40} T^{20} - \)\(17\!\cdots\!12\)\( p^{60} T^{21} - \)\(54\!\cdots\!12\)\( p^{80} T^{22} - \)\(12\!\cdots\!12\)\( p^{100} T^{23} + \)\(22\!\cdots\!72\)\( p^{120} T^{24} + \)\(61\!\cdots\!56\)\( p^{140} T^{25} + \)\(94\!\cdots\!68\)\( p^{161} T^{26} + \)\(44\!\cdots\!56\)\( p^{180} T^{27} + \)\(23\!\cdots\!20\)\( p^{200} T^{28} + \)\(11\!\cdots\!84\)\( p^{220} T^{29} + \)\(64\!\cdots\!84\)\( p^{240} T^{30} + \)\(31\!\cdots\!84\)\( p^{260} T^{31} + \)\(11\!\cdots\!93\)\( p^{280} T^{32} + \)\(36\!\cdots\!52\)\( p^{300} T^{33} + \)\(15\!\cdots\!52\)\( p^{320} T^{34} + 5597336519263153752 p^{340} T^{35} + p^{360} T^{36} \) | |
| 71 | \( ( 1 - 5491638747787112368 T + \)\(56\!\cdots\!53\)\( T^{2} - \)\(22\!\cdots\!16\)\( T^{3} + \)\(13\!\cdots\!60\)\( T^{4} - \)\(51\!\cdots\!68\)\( T^{5} + \)\(23\!\cdots\!64\)\( T^{6} - \)\(84\!\cdots\!24\)\( T^{7} + \)\(31\!\cdots\!58\)\( T^{8} - \)\(10\!\cdots\!20\)\( T^{9} + \)\(31\!\cdots\!58\)\( p^{20} T^{10} - \)\(84\!\cdots\!24\)\( p^{40} T^{11} + \)\(23\!\cdots\!64\)\( p^{60} T^{12} - \)\(51\!\cdots\!68\)\( p^{80} T^{13} + \)\(13\!\cdots\!60\)\( p^{100} T^{14} - \)\(22\!\cdots\!16\)\( p^{120} T^{15} + \)\(56\!\cdots\!53\)\( p^{140} T^{16} - 5491638747787112368 p^{160} T^{17} + p^{180} T^{18} )^{2} \) | |
| 73 | \( 1 + 19072400700441139902 T + \)\(18\!\cdots\!02\)\( T^{2} + \)\(11\!\cdots\!02\)\( T^{3} + \)\(66\!\cdots\!41\)\( p T^{4} + \)\(14\!\cdots\!08\)\( p T^{5} + \)\(24\!\cdots\!84\)\( T^{6} + \)\(22\!\cdots\!84\)\( T^{7} + \)\(13\!\cdots\!20\)\( T^{8} + \)\(12\!\cdots\!56\)\( T^{9} + \)\(56\!\cdots\!56\)\( T^{10} + \)\(14\!\cdots\!56\)\( T^{11} + \)\(68\!\cdots\!72\)\( T^{12} - \)\(19\!\cdots\!12\)\( T^{13} + \)\(63\!\cdots\!88\)\( T^{14} + \)\(61\!\cdots\!88\)\( T^{15} + \)\(28\!\cdots\!14\)\( T^{16} + \)\(80\!\cdots\!40\)\( T^{17} + \)\(23\!\cdots\!40\)\( T^{18} + \)\(80\!\cdots\!40\)\( p^{20} T^{19} + \)\(28\!\cdots\!14\)\( p^{40} T^{20} + \)\(61\!\cdots\!88\)\( p^{60} T^{21} + \)\(63\!\cdots\!88\)\( p^{80} T^{22} - \)\(19\!\cdots\!12\)\( p^{100} T^{23} + \)\(68\!\cdots\!72\)\( p^{120} T^{24} + \)\(14\!\cdots\!56\)\( p^{140} T^{25} + \)\(56\!\cdots\!56\)\( p^{160} T^{26} + \)\(12\!\cdots\!56\)\( p^{180} T^{27} + \)\(13\!\cdots\!20\)\( p^{200} T^{28} + \)\(22\!\cdots\!84\)\( p^{220} T^{29} + \)\(24\!\cdots\!84\)\( p^{240} T^{30} + \)\(14\!\cdots\!08\)\( p^{261} T^{31} + \)\(66\!\cdots\!41\)\( p^{281} T^{32} + \)\(11\!\cdots\!02\)\( p^{300} T^{33} + \)\(18\!\cdots\!02\)\( p^{320} T^{34} + 19072400700441139902 p^{340} T^{35} + p^{360} T^{36} \) | |
| 79 | \( 1 - \)\(40\!\cdots\!18\)\( T^{2} + \)\(99\!\cdots\!53\)\( T^{4} - \)\(18\!\cdots\!16\)\( T^{6} + \)\(27\!\cdots\!60\)\( T^{8} - \)\(35\!\cdots\!68\)\( T^{10} + \)\(41\!\cdots\!64\)\( T^{12} - \)\(45\!\cdots\!24\)\( T^{14} + \)\(45\!\cdots\!58\)\( T^{16} - \)\(42\!\cdots\!20\)\( T^{18} + \)\(45\!\cdots\!58\)\( p^{40} T^{20} - \)\(45\!\cdots\!24\)\( p^{80} T^{22} + \)\(41\!\cdots\!64\)\( p^{120} T^{24} - \)\(35\!\cdots\!68\)\( p^{160} T^{26} + \)\(27\!\cdots\!60\)\( p^{200} T^{28} - \)\(18\!\cdots\!16\)\( p^{240} T^{30} + \)\(99\!\cdots\!53\)\( p^{280} T^{32} - \)\(40\!\cdots\!18\)\( p^{320} T^{34} + p^{360} T^{36} \) | |
| 83 | \( 1 + \)\(11\!\cdots\!52\)\( T + \)\(71\!\cdots\!52\)\( T^{2} + \)\(29\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!71\)\( p T^{4} + \)\(27\!\cdots\!84\)\( T^{5} + \)\(69\!\cdots\!84\)\( T^{6} + \)\(16\!\cdots\!84\)\( T^{7} + \)\(35\!\cdots\!20\)\( T^{8} + \)\(73\!\cdots\!56\)\( T^{9} + \)\(14\!\cdots\!56\)\( T^{10} + \)\(27\!\cdots\!56\)\( T^{11} + \)\(51\!\cdots\!72\)\( T^{12} + \)\(92\!\cdots\!88\)\( T^{13} + \)\(16\!\cdots\!88\)\( T^{14} + \)\(27\!\cdots\!88\)\( T^{15} + \)\(45\!\cdots\!14\)\( T^{16} + \)\(73\!\cdots\!40\)\( T^{17} + \)\(11\!\cdots\!40\)\( T^{18} + \)\(73\!\cdots\!40\)\( p^{20} T^{19} + \)\(45\!\cdots\!14\)\( p^{40} T^{20} + \)\(27\!\cdots\!88\)\( p^{60} T^{21} + \)\(16\!\cdots\!88\)\( p^{80} T^{22} + \)\(92\!\cdots\!88\)\( p^{100} T^{23} + \)\(51\!\cdots\!72\)\( p^{120} T^{24} + \)\(27\!\cdots\!56\)\( p^{140} T^{25} + \)\(14\!\cdots\!56\)\( p^{160} T^{26} + \)\(73\!\cdots\!56\)\( p^{180} T^{27} + \)\(35\!\cdots\!20\)\( p^{200} T^{28} + \)\(16\!\cdots\!84\)\( p^{220} T^{29} + \)\(69\!\cdots\!84\)\( p^{240} T^{30} + \)\(27\!\cdots\!84\)\( p^{260} T^{31} + \)\(11\!\cdots\!71\)\( p^{281} T^{32} + \)\(29\!\cdots\!52\)\( p^{300} T^{33} + \)\(71\!\cdots\!52\)\( p^{320} T^{34} + \)\(11\!\cdots\!52\)\( p^{340} T^{35} + p^{360} T^{36} \) | |
| 89 | \( 1 - \)\(99\!\cdots\!18\)\( T^{2} + \)\(50\!\cdots\!53\)\( T^{4} - \)\(17\!\cdots\!16\)\( T^{6} + \)\(44\!\cdots\!60\)\( T^{8} - \)\(90\!\cdots\!68\)\( T^{10} + \)\(15\!\cdots\!64\)\( T^{12} - \)\(21\!\cdots\!24\)\( T^{14} + \)\(26\!\cdots\!58\)\( T^{16} - \)\(27\!\cdots\!20\)\( T^{18} + \)\(26\!\cdots\!58\)\( p^{40} T^{20} - \)\(21\!\cdots\!24\)\( p^{80} T^{22} + \)\(15\!\cdots\!64\)\( p^{120} T^{24} - \)\(90\!\cdots\!68\)\( p^{160} T^{26} + \)\(44\!\cdots\!60\)\( p^{200} T^{28} - \)\(17\!\cdots\!16\)\( p^{240} T^{30} + \)\(50\!\cdots\!53\)\( p^{280} T^{32} - \)\(99\!\cdots\!18\)\( p^{320} T^{34} + p^{360} T^{36} \) | |
| 97 | \( 1 + \)\(31\!\cdots\!02\)\( T + \)\(50\!\cdots\!02\)\( T^{2} + \)\(48\!\cdots\!02\)\( T^{3} + \)\(34\!\cdots\!93\)\( T^{4} + \)\(27\!\cdots\!84\)\( T^{5} + \)\(31\!\cdots\!84\)\( T^{6} + \)\(30\!\cdots\!84\)\( T^{7} + \)\(21\!\cdots\!20\)\( T^{8} + \)\(13\!\cdots\!56\)\( T^{9} + \)\(14\!\cdots\!56\)\( T^{10} + \)\(15\!\cdots\!56\)\( T^{11} + \)\(12\!\cdots\!72\)\( T^{12} + \)\(70\!\cdots\!88\)\( T^{13} + \)\(53\!\cdots\!88\)\( T^{14} + \)\(54\!\cdots\!88\)\( T^{15} + \)\(46\!\cdots\!14\)\( T^{16} + \)\(27\!\cdots\!40\)\( T^{17} + \)\(16\!\cdots\!40\)\( T^{18} + \)\(27\!\cdots\!40\)\( p^{20} T^{19} + \)\(46\!\cdots\!14\)\( p^{40} T^{20} + \)\(54\!\cdots\!88\)\( p^{60} T^{21} + \)\(53\!\cdots\!88\)\( p^{80} T^{22} + \)\(70\!\cdots\!88\)\( p^{100} T^{23} + \)\(12\!\cdots\!72\)\( p^{120} T^{24} + \)\(15\!\cdots\!56\)\( p^{140} T^{25} + \)\(14\!\cdots\!56\)\( p^{160} T^{26} + \)\(13\!\cdots\!56\)\( p^{180} T^{27} + \)\(21\!\cdots\!20\)\( p^{200} T^{28} + \)\(30\!\cdots\!84\)\( p^{220} T^{29} + \)\(31\!\cdots\!84\)\( p^{240} T^{30} + \)\(27\!\cdots\!84\)\( p^{260} T^{31} + \)\(34\!\cdots\!93\)\( p^{280} T^{32} + \)\(48\!\cdots\!02\)\( p^{300} T^{33} + \)\(50\!\cdots\!02\)\( p^{320} T^{34} + \)\(31\!\cdots\!02\)\( p^{340} T^{35} + p^{360} T^{36} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.18979058826917531944207293405, −3.18076165367505940583305706662, −2.98994724394567330638011177770, −2.92328865670922642446819002790, −2.83056210102799081122086312529, −2.80260063120540767602143525210, −2.74273667840353159051587413981, −2.43207447112069520842769232421, −2.31428979255738228090114988708, −2.26066441428551692386591223093, −1.81168490950033558138649206795, −1.77561223380849530413802737923, −1.67317257598719022455065444383, −1.59242723583644633540065588216, −1.39058026553693245579315227640, −1.35579833420532458318197378637, −1.33737886521751817514683045306, −1.05499248837011750081026276632, −0.837383946036322890799224378024, −0.72553998038403779295825759939, −0.46176595778694518772488728045, −0.43026022227495722232061114622, −0.22892235106816387724810329527, −0.14531743132646571948372583694, −0.03582136484396223408656006151, 0.03582136484396223408656006151, 0.14531743132646571948372583694, 0.22892235106816387724810329527, 0.43026022227495722232061114622, 0.46176595778694518772488728045, 0.72553998038403779295825759939, 0.837383946036322890799224378024, 1.05499248837011750081026276632, 1.33737886521751817514683045306, 1.35579833420532458318197378637, 1.39058026553693245579315227640, 1.59242723583644633540065588216, 1.67317257598719022455065444383, 1.77561223380849530413802737923, 1.81168490950033558138649206795, 2.26066441428551692386591223093, 2.31428979255738228090114988708, 2.43207447112069520842769232421, 2.74273667840353159051587413981, 2.80260063120540767602143525210, 2.83056210102799081122086312529, 2.92328865670922642446819002790, 2.98994724394567330638011177770, 3.18076165367505940583305706662, 3.18979058826917531944207293405
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.